Is a Simply Connected Space Homotopically equivalent to a point

Question

If X is a simply connected Space is it homotopically equivalent to a point in the space, I know this holds in $ \mathbb{R}^n $ but only because of its algebraic properties does it hold for general topologies?

Answer 1

A sphere of dimension 2 is simply connected and not equivalent to a point.

A space homotopically equivalent to a point not only has its fundamental group trivial but also all its higher homotopy groups. There is a famous theorem of Whitehead that says that, provided the space is sufficiently good, having all homotopy groups trivial does imply contractibility; sufficiently good means, for example, being a CW-complex.